Theorem opid 3646

Description: The ordered pair ⟨𝐴, 𝐴⟩ in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Hypothesis

Ref	Expression
opid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
opid	⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Proof of Theorem opid

Step	Hyp	Ref	Expression
1		dfsn2 3464	. . . 4 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	eqcomi 2093	. . 3 ⊢ {𝐴, 𝐴} = {𝐴}
3	2	preq2i 3527	. 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}}
4		opid.1	. . 3 ⊢ 𝐴 ∈ V
5	4, 4	dfop 3627	. 2 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}, {𝐴, 𝐴}}
6		dfsn2 3464	. 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}}
7	3, 5, 6	3eqtr4i 2119	1 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}

Syntax hints:   = wceq 1290   ∈ wcel 1439  Vcvv 2620  {csn 3450  {cpr 3451  ⟨cop 3453

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459

This theorem is referenced by:  dmsnsnsng  4921  funopg  5061